3 Matrix-monotone functions.

The last chapter discussed a certain partial order for vectors Majorization end characterized the order preserving functions Schur-convex and Schur-concave functions. This chapter will propose a certain order on matrices, namely the Lowner order. The order preserving functions are called matrix-monotone functions and their representations and properties will be discussed.

3.1 Definition and Examples

To define monotonicity for matrix-valued functions of matrices, a partial order on the set of all n x n positive semidefnite matrices is required. Define the set of all positive semidefnite matrices of size n x n as [H.sub.n]. One standard ordering which is called the Lowner ordering is described by

A [less than or equal to] B means B - A is positive semidefnite

A < B means B - A is positive definite.

In order to transfer the notion of monotonicity and convexity to matrices and matrix-valued functions, the following definitions are helpful.

Let the function [phi] be an arbitrary matrix-valued function. The function [phi] maps from the set of positive semidefnite matrices to the set of positive semidefnite matrices. The function 0 is defined in the following [8, 51].

Definition 3.1 (Matrix function). Let the eigenvalue decomposition of A be given as A = U[LAMBDA][U.sup.H]. Then [phi](A) means

[phi](A) = U[phi]([LAMBDA]) [U.sup.H] with [phi]([LAMBDA]) = diag ([phi]([[lambda].sub.1]]), ..., [phi]([[lambda].sub.n])).

That means, the function [phi] affects only the eigenvalues of the matrix A and keeps the eigenvectors unaltered.

Remark 3.1. There are many possible definitions of how a function affects a matrix. The Definition 3.1 is motivated by the Spectral Theorem (Proposition 6.2 in Section 6.1). It follows that the matrix function [phi] can be regarded as scalar function f : [R.sup.+] [right arrow] [R.sup.+]. The interested reader is referred to Chapter 6 of [51].

3.1.1 Matrix Monotonicity

The concept of matrix-monotonicity was fist studied by [88].

Definition 3.2 (Matrix-increasing). Let A [subset] [H.sub.n]. A function [phi] A [right arrow] [H.sub.n] is matrix-increasing of order n on A if

A [less than or equal to] B implies [phi](A) [less than or equal to] [phi](B) (3.1)

for A, B [member of] A. The function is strictly matrix-increasing of order n on A if

A < B implies [phi](A) < [phi](B)

Remark 3.2. If a function is matrix-increasing for all orders n [greater than or equal to] 1 it is just called matrix-increasing or sometimes also matrix-monotone.

Recently, it was shown that the class of matrix-monotone functions of order n + 1 is a proper subset of the class of matrix-monotone functions of order n [45].

Let us study some examples and counter examples of matrix-monotonicity.

Lemma 3.1 (16.E.3.b in [92]). On the set of positive definite matrices, the function [phi](A) = [A.sup.-1] is strictly decreasing.

Proof. Let [Q.sub.1] [greater than or equal to] [Q.sub.2] > 0 and define

g(t) = (t[Q.sub.1] + [(1 - t)[Q.sub.2]).sup.-1] = Q[(t).sup.-1].

It suffices to prove that g is strictly decreasing in 0 [less than or equal to] t [less than or equal to] 1. Note that Q(t)Q[(t).sup.-1] = I. As a result

[partial derivative]Q(t)/[partial derivative]t [Q.sup.-1](t) + Q(t) [partial derivative]Q[(t).sup.-1]/[partial derivative]t = 0.

Hence,

[partial derivative]Q[(t).sup.-1]/[partial derivative]t = -Q[(t).sup.-1] ([partial derivative]Q(t)/[partial derivative]t) Q[(t).sup.-1] = -Q[(t).sup.-1] ([Q.sub.1] - [Q.sub.2])Q[(t).sup.-1] < 0.

Example 3.1. On the set of positive definite matrices, the function [phi](A) = [A.sup.2] is not strictly increasing. Consider the following counter example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The eigenvalues of the matrix [A.sup.2] - [B.sup.2] are -10.94626578 and 993.9462658. The example is a special case of [92, 16.E.4] and it shows that conclusions from the scalar case to the matrix case have to be drawn very carefully.

3.1.2 Matrix Convexity

The concept of matrix-convexity was fist studied by [83].

Definition 3.3 (Matrix-convex). Let A [subset] [H.sub.n]. A function [phi]: A [right arrow] [H.sub.n], is matrix-convex of order n on A if

[phi]([alpha]A + [??]B) [less than or equal to] [alpha](A) + [??][phi](B) for all [alpha][member of][0, 1] and A, B [member of] A (3.2)

[phi] is strictly matrix-convex of order n on A if

[phi]([alpha]A + [??]B) < [??][phi](A) + [??][phi](B) for all [alpha] [member of] [0,1] and A, B [member of] A.

A function is called just matrix-convex if it is matrix-convex for all orders n [greater than or equal to] 1.

Theorem 3.2 (Theorem 2.5 in [46]). A nonnegative continuous function on [0, [infinity]) is operator monotone if and only if it is operator concave.

Remark 3.3. However, not every matrix-convex function is necessarily matrix-monotone. The function [phi](A) = [A.sup.2] is matrix-convex but not matrix-monotone as shown in Example 3.1.

Proposition 3.3 (Proposition 16.E.6 in [92]). Let [phi] be a function defined on a convex set A of m x k matrices, taking values in [H.sub.n], for some n. If A is open and g is twice differentiable for all A, B [member of] A the following are equivalent:

(1) [phi] is matrix-convex on A.

(2) For all fixed A and B in A, the function g(a) = [phi]([alpha]A + [??]B) is convex in a [member of] [0,1] in the sense that [eta]([alpha]) + [??]g([beta]) - G([eta][alpha] + [??]B) is positive semidefinite for all [alpha], [beta], [eta] [member of] [0,1].

(3) For all fixed A, B [member of] A, [d.sup.2]g([alpha])/d[[alpha].sup.2] is positive semidefinite for 0 < [alpha] < 1.

Proof. Matrix convexity on A is fulfilled if and only if for all x [member of] [R.sup.n], [psi](A) = x[phi](A)[x.sup.H] is convex. [psi](A) is convex on A if and only if [psi]"(A) = x [d.sup.2]g([alpha])/d[[alpha].sup.2] [x.sup.H] [greater than or equal to] 0 for all A [member of] A.

The next example illustrates the notion of matrix-concavity.

Example 3.2. Consider f (X) = log(I + X). In order to show that f is matrix-concave, we need the following identity:

[d.sup.2]log(I+[alpha]X)/d[[alpha].sup.2] = -X[[I + [alpha]X].sup.-2] X [less than or equal to] 0.

As a result, for g([alpha]) = log(I + [alpha]X + [??]Y) it holds

[d.sup.2]g([alpha])/d[[alpha].sup.2] [less than or equal to] 0

matrix-convexity follows by Proposition 3.3. Matrix-monotonicity follows from Theorem 3.2.

The properties of the matrix function 0 can be transferred to scalar function 0 but not vice versa.

Corollary 3.1. Every matrix-monotone function is monotonic (increasing or decreasing) whereas not every monotonic function is matrix monotone. Every matrix convex function is convex whereas not every convex function is matrix convex.

Example 3.3. The function [phi](x) = [x.sup.2] is increasing and convex but not matrix monotone. The function [phi](x) = exp(x) is convex but not matrix convex.

The connection between the scalar properties of monotonicity and concavity and the matrix-monotonicity is illustrated in the Venn-diagram in Figure 3.1.

[FIGURE 3.1 OMITTED]

3.1.3 Frechet Derivative

Corresponding to the fist and second derivatives of scalar functions, there exists a derivative of the matrix-valued function [phi]. We follow closely the derivation in [8, Sec. V.3 and Sec. X.4].

Definition 3.4 (Frechet differentiable). The map [phi] is called (Frechet) differentiable at A if there exists a linear transformation D[phi](A) on the space of positive semidefinite matrices such that for all H

[parallel][phi](A + H) - [phi](A) - D[phi](A) (H)[parallel] = o([parallel]H[parallel]) (3.3)

holds. The linear operator D[alpha](A)(H) is then called the derivative of [alpha] at A in direction H.

The difference from the scalar case is that a direction H is needed to define the derivative.

Lemma 3.4 (First derivative in a direction). The fist derivative of [phi](A) = [A].sup.p] in direction of B is given by

D[phi](A)(B) = [p.summation over (k=1)] [A.sup.k-1]B[A.sup.p-k].

Proof. Compute the first derivative of [phi]([epsilon]) = [(A + [member of]B).sup.p] with respect to [epsilon] at the point [epsilon] = 0. Using the product rule, it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At the point [epsilon] = 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof.

Example 3.4. Next, three examples are provided to illustrate the notion of the derivative of a matrix function.

(1) Let [phi](A) = [A.sup.2]. Then

D[phi](A) (B) = AB + BA.

(2) Let [phi](A) = [A.sup.-1] for invertible A. Then

D[phi](A)(B) = [-A.sup.-1]B[A.sup.-1].

(3) Let [phi](A) = [A.sup.H]A. Then

D[phi](A) (B) = [A.sup.H] B + [B.sup.H] A.

The derivative is linear, i.e., D([[phi].sub.1] + [[phi].sub.2])(A) = D([[phi].sub.1](A)) + D([[phi].sub.2](A)).

The composite of two differentiable maps [phi] and [gamma] is differentiable.

The chain rule is D([gamma] x [phi])(A) = D[gamma]([phi](A)) x D[phi](A). Interestingly, if the trace operator is applied to the function [phi], the following result can be proven. This is a special case of the results in [471.

Lemma 3.5 (Derivative of trace of matrix function). Let [phi] be a continuous analytic function mapping positive semidefinite matrices to positive semidefinite matrices. Then, the fist derivative of the function tr [[phi](C + [epsilon]D)] with respect to [epsilon] at the point [member of] = 0 is given by